Snapping Jumps to wrong place
You're after the Snapping menu.  When snapping is turned on, the magnet is solid with a wavy line, and you can configure what feature the snapping applies to.
Snapping off:

Snapping on:

To turn off snapping mid-transform, use Shift + Tab:

See the snapping documentation here: https://docs.blender.org/manual/en/2.79/editors/3dview/object/editing/transform/control/snap.html